Categories
Singapore Math Olympiad

Functional Equation Problem from SMO, 2018 – Question 35

Try this problem from Singapore Mathematics Olympiad, SMO, 2018 based on Functional Equation. You may use sequential hints if required.

Try to solve this problem number 35 from Singapore Mathematics Olympiad, SMO, 2018 based on Functional Equation.

Problem – Functional Equation (SMO Entrance)


Consider integers \({1,2, \ldots, 10}\). A particle is initially -at 1 . It moves to an adjacent integer in the next step. What is the expected number of steps it will take to reach 10 for the first time?

  • 82
  • 81
  • 80
  • 79

Key Concepts


Functional Equation

Equation

Check the Answer


Answer : 81

Singapore Mathematical Olympiad

Challenges an Thrills – Pre – College Mathematics

Try with Hints


If you got stuck into this problem we can start taking an expected number of steps to be \(g_{n}\). We need to remember at first the particle was in 1 then it will shift to the next step so for n no of position we can expressed it as n and n -1 where n = 2,3,4,……..,100.

Now try the rest…………..

Now let’s continue after the last hint …………

Then \(g_{n+1} = \frac {1}{2} (1+g_{n} + g_{n+1} )+ \frac {1}{2}\)

which implies , \(g_{n+1} = g_{n} + 2\)

Now we know that,\(g_{2} = 1\). Then \(g_{3} = 3\), \(g_{4}= 5\),………………,\(g_{10}=17\)

\(g = g_{2}+g_{3}+g_{4}+………………..+g_{10} = 1+3+…………………+17 = 81\)[ Answer]

Subscribe to Cheenta at Youtube


Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.